1.2 Tidal problem around a BH: Effects of BH spin and the NS EOS

As shown by a simple estimate in the previous Section 1.1, the final fate of BH-NS binaries depends
primarily on the mass ratio, , and the compactness of the NS, . However, a detailed analysis has
shown that the BH spin and the NS EOS also play an important role in determining the final fate. In
particular, the effect of the BH spin can be significant.

A general relativistic study for the criterion of mass shedding (note again that this is not the criterion of
tidal disruption) was first performed using an analysis of the tidal interaction between a fluid star and a
Kerr BH in circular orbit [71, 137, 135, 191, 225, 94, 69, 70, 154]. In this class of analysis, one handles a
fluid star orbiting a Kerr BH in a circular orbit (the center of the fluid star has a geodesic motion around
the Kerr BH), and analyzes the structure of the star taking into account the tidal field of the
Kerr BH located at the center. Specifically, one analyzes the Euler equation in the following form

where is the affine parameter of a geodesic around the BH, is a coordinate orthogonal to the
geodesic, denotes the internal velocity of the fluid star, is the rest-mass density, is the
pressure, is the Newtonian potential, which obeys , and determines the lowest-order
tidal field [135]. Higher-order corrections of the tidal potential based on the Manasse–Misner
formalism [134] are also given in [94].

The gravitational effect of the fluid star on the orbital motion, the gravitational radiation reaction, and
the relativistic effect on the self-gravity of the star are in general neglected (see [69] for a more
detailed phenomenological analysis). Thus, this analysis is valid only for the case in which the BHmass is much larger than the NS mass and t the radius of the NS is relatively large: For the
case of BH-NS binaries of mass ratio 10, this analysis can provide only a qualitative or
semi-quantitative nature of the tidal effect and orbital evolution. However, by this analysis, several
important qualitative properties of the criterion of mass shedding have been found. One of the
most important findings is that the criteria for the onset of mass shedding (and/or for tidal
disruption) depend sensitively on the BH spin, . The reason for this is that the angular
velocity at the ISCO depends sensitively on the BH spin; for the
non-spinning BH whereas it is 1/2 for the BH of maximum possible spin, [16, 186]. Here,
the spin is non-dimensional and defined by where is the angular
momentum of the BH. As Equation (10) indicates, the increase of results in the increase
of the critical value of for mass shedding (tidal disruption). Indeed, analyses [191, 225]
indicate that the critical value for (or BH mass) increases by a factor of 15 if the
spin is changed from zero to the maximum value (); e.g., for an incompressible fluid
star, the maximum possible mass of a BH, which can cause mass shedding, was derived as

where for for for for , and for
.

Another important finding is that the criteria for mass shedding (and tidal disruption) depend on the
NS EOS even if the mass and radius of the NS are identical [225, 94]: NSs with stiffer EOS, (i.e., with high
adiabatic index) have a relatively uniform density profile and are susceptible to tidal deformation, mass
shedding, and tidal disruption by the BH tidal field. Consequently, condition (12) is modified by the EOS.
In the calculation with compressible models [225, 94], the maximum possible mass of a BH,
which can cause mass shedding, may be reduced by 10 – 20% for soft EOS (with the relatively
small adiabatic index). This implies that not only the compactness of the NS but also its EOS,
which is still unknown, is reflected in the tidal disruption event. General relativistic studies for
quasi-equilibria of BH-NS binaries in addition show that the self-gravity of the NS significantly
reduces the maximum mass of the BH for the onset of mass shedding (see Section 1.4 and
Figure 22).